In this work we suggest a new numerical method to treat nucleation and cavitation or more generally two-phase flow problems with sharp interfaces that result in the existence of tiny cells. We treat the neighborhood of the tiny cells implicitly for stability while using an explicit time stepping scheme everywhere else to keep the cost low. Using flux bounding to couple the schemes as introduced in Ref. [1]} results in the mixed explicit implicit scheme being conservative and stable. We suggest to use dual time stepping for solving the resulting implicit systems in the neighborhood of the tiny cells.
Our numerical results, which include nucleation and cavitation tests, show an accurate and robust performance of the mixed scheme. In terms of running times we observed significant speed-ups compared to a fully explicit local time stepping approach.
In the future, we plan to explore the speed-up options further by looking into better starting points for the dual time stepping (instead of simply using \(\mathbf {U}^n\) ) or Newton-like approaches. We also plan to extend the approach to two dimensions.
